Persistence of Hyperbolic-type Degenerate Lower-dimensional Invariant Tori with Prescribed Frequencies in Hamiltonian Systems

It is known that under Kolmogorov's nondegeneracy condition, the nondegenerate hyperbolic invariant torus with Diophantine frequencies will persist under small perturbations, meaning that the perturbed system still has an invariant torus with prescribed frequencies. However, the degenerate torus is sensitive to perturbations. In this paper, we prove the persistence of two classes of hyperbolic-type degenerate lower-dimensional invariant tori, one of them corrects an earlier work [34] by the second author. The proof is based on a modified KAM iteration and analysis of stability of degenerate critical points of analytic functions.